Angles & Basic Geometry Review

Administrative Announcements

• “Coffee-session” group reviews
  • Max 12 students per batch; slots released via Google Sheet next Tuesday – first-come, first-served.
  • Schedule: Thursday or Friday, 2 PM & 7 PM (handled only by Ms. Jessie).
  • Separate Davao sessions start next week (teacher: Sir Dyron).
• Review coverage priorities for college entrance tests (UPCAT, UCAT, DCAT, etc.)
  • Geometry appears late in the UPCAT (Items 35-45), usually ≤10 pts.
  • Heaviest UPCAT topics: Algebra, General Math (Grade 11), Triangle & Circle problems.
  • Typical higher-level questions involve trig identities or exact values (e.g., $(\sin 130^\circ)$).

Foundational Vocabulary & Notation

• Angle
  • Formed by rotating a ray from an initial side to a terminal side about a common vertex.
  • Symbol: $m\angle ABC$ means “measure of angle ABC.”
• Initial Side vs. Terminal Side
  • Initial: the fixed ray before rotation.
  • Terminal: the position after rotation.
• Positive/Negative Direction
  • Counter-clockwise rotation → positive angle.
  • Clockwise rotation → negative angle.

Measuring Angles

• Degree measure
  • Defined as a fraction of one full revolution: $1^\circ = \frac{1}{360}\text{ of a turn}$.
• Radian measure
  • $\pi\text{ rad} = 180^\circ$ ; therefore $1\text{ rad} = \frac{180^\circ}{\pi}$.
  • Full revolution $= 2\pi\text{ rad}=360^\circ$.
• Percentage idea: “Magnitude” of an angle is its fraction of $360^\circ$.

Point (Circular) Definition of Trig Functions

• For a point $P(x,y)$ on the unit circle
  • $\cos\theta = x$ (horizontal coordinate).
  • $\sin\theta = y$ (vertical coordinate).
• A line through $P$ tangent to the circle defines $\tan\theta$.
• “Point functions” = trigonometric functions (cos x, sin x, etc.).

Reference Trig Values & Cofunction Identities

• Special exact values
  • $\tan 30^\circ = \cot 60^\circ = \frac{\sqrt{3}}{3}$.
• Cofunction (complementary) identities
  • $\sin\theta = \cos(90^\circ-\theta)$
  • $\tan\theta = \cot(90^\circ-\theta)$
  • $\sec\theta = \csc(90^\circ-\theta)$.

Classification of Angles

• Acute: 0^\circ < \theta < 90^\circ (e.g., $45^\circ$).
• Right: $\theta = 90^\circ$.
• Obtuse: 90^\circ < \theta < 180^\circ (e.g., $179.9^\circ$).
• Straight: $\theta = 180^\circ$ (forms a straight line).
• Reflex: 180^\circ < \theta < 360^\circ.
• Complete (Full/Perigon): $\theta = 360^\circ$.
• Zero angle: $\theta = 0^\circ$ (initial and terminal sides coincide).

Angle Addition & Interior-Point Postulate

• If point $D$ lies in the interior of $\angle ABC$ then
  $m\angle ABD + m\angle DBC = m\angle ABC.$
• Example: If $m\angle ABD = x$ and $m\angle DBC = 50^\circ$ then $m\angle ABC = x+50^\circ$; conversely $m\angle ABD = x-50^\circ.$

Complementary & Supplementary Angles

• Complementary → sum $=90^\circ$.
  • Complement of $x$ is $90^\circ - x$.
• Supplementary → sum $=180^\circ$.
  • Supplement of $x$ is $180^\circ - x$.
• Sample Word-Problem (from lecture)
  • “An angle is $18^\circ$ less than twice its complement.”
  • Let angle $=x$. Equation: $x = 2(90 - x) - 18$ → $x=54^\circ$ (complement $=36^\circ$).

Special Angle Relationships at an Intersection

• Adjacent angles
  • Share a common vertex and a common side; non-common sides lie on opposite sides of the shared side.
• Linear pair (straight angle)
  • Two adjacent angles forming $180^\circ$.
• Vertical (opposite) angles
  • Non-adjacent angles formed by two intersecting lines; always congruent.
  • Example: If one vertical angle $=5x^\circ$ and its linear partner $=(2x-30)^\circ$, solve $5x + 2x - 30 = 180$ → $x=30$.

Perpendicular & Parallel Lines

• Perpendicular lines intersect forming $90^\circ$ at all four corners.
• Parallel lines: never meet; technically, “two lines in the same plane that maintain a constant distance and have no intersection.”
  • Must be explicitly stated – do NOT assume parallelism from a textbook sketch.

Transversals & Angle Pairs

Given two (stated) parallel lines cut by a transversal:

• Corresponding angles (same relative position) are congruent.
  • Examples: $\angle1 \cong \angle5,\; \angle2 \cong \angle6,\; \angle3 \cong \angle7,\; \angle4 \cong \angle8$.
• Alternate Interior Angles (inside, opposite sides of transversal) are congruent.
  • $\angle3 \cong \angle6,\; \angle4 \cong \angle5$.
• Alternate Exterior Angles (outside, opposite sides) are congruent.
  • $\angle1 \cong \angle8,\; \angle2 \cong \angle7$.
• Same-Side (Consecutive) Interior Angles are supplementary.
  • $m\angle3 + m\angle5 = 180^\circ,\; m\angle4 + m\angle6 = 180^\circ.$
• Same-Side Exterior Angles are also supplementary.
  • $m\angle1 + m\angle7 = 180^\circ,\; m\angle2 + m\angle8 = 180^\circ.$
• Example Calculation: If $m\angle1 = 50^\circ$ then $m\angle7 = 130^\circ$ by same-side exterior rule.

Polygon Interior-Angle Sum (mentioned briefly)

• General formula: $S = (n-2)\times180^\circ$.
  • For a pentagon ($n=5$) ⇒ $S = 540^\circ$.

Exam-Focused Reminders

• Geometry items typically modest in number; prioritize accuracy over speed.
• Know special trig exact values and cofunction identities for short-answer questions.
• Practice solving word problems using “less than,” “twice,” etc., translating words to equations correctly (keyword: order reversal for “less than”).

Real-World & Cross-Disciplinary Connections

• Color complement analogy: blue vs. orange light absorption/reflection.
• Hyperbolic trig (“sinh,” “cosh”) briefly noted as arising from hyperbola analog to unit circle.

Quick Problem-Solving Checklist

• Identify given relationships (vertical, linear pair, corresponding, etc.).
• Mark all right angles and straight lines immediately.
• Write algebraic equations using $m\angle$ notation; isolate variables.
• Verify if “parallel” is explicitly stated before applying transversal theorems.